Theor. Comput. Fluid Dyn.DOI 10.1007/s00162-010-0184-8

ORIGINAL ARTICLE

Zhanhua Ma · Sunil Ahuja · Clarence W. Rowley

Reduced-order models for control of fluidsusing the eigensystem realization algorithm

Received: 23 June 2009 / Accepted: 19 November 2009© Springer-Verlag 2010

Abstract As sensors and flow control actuators become smaller, cheaper, and more pervasive, the use offeedback control to manipulate the details of fluid flows becomes increasingly attractive. One of the challengesis to develop mathematical models that describe the fluid physics relevant to the task at hand, while neglect-ing irrelevant details of the flow in order to remain computationally tractable. A number of techniques arepresently used to develop such reduced-order models, such as proper orthogonal decomposition (POD), andapproximate snapshot-based balanced truncation, also known as balanced POD. Each method has its strengthsand weaknesses: for instance, POD models can behave unpredictably and perform poorly, but they can becomputed directly from experimental data; approximate balanced truncation often produces vastly superiormodels to POD, but requires data from adjoint simulations, and thus cannot be applied to experimental data.In this article, we show that using the Eigensystem Realization Algorithm (ERA) (Juang and Pappa, J GuidControl Dyn 8(5):620–627, 1985) one can theoretically obtain exactly the same reduced-order models as bybalanced POD. Moreover, the models can be obtained directly from experimental data, without the use ofadjoint information. The algorithm can also substantially improve computational efficiency when formingreduced-order models from simulation data. If adjoint information is available, then balanced POD has someadvantages over ERA: for instance, it produces modes that are useful for multiple purposes, and the methodhas been generalized to unstable systems. We also present a modified ERA procedure that produces modeswithout adjoint information, but for this procedure, the resulting models are not balanced, and do not performas well in examples. We present a detailed comparison of the methods, and illustrate them on an example ofthe flow past an inclined flat plate at a low Reynolds number.

Keywords Flow control · Model reduction · Eigensystem Realization Algorithm · Balanced truncation

1 Introduction

In the last decade, substantial developments have been made in the area of model-based feedback flow controlof fluids: for instance, see the recent reviews by [7,8,17]. In many applications, the focus is on how to apply

Communicated by T. Colonius

Z. Ma (B), S. Ahuja, C. W. RowleyDepartment of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USAE-mail: zma@princeton.edu

S. AhujaE-mail: sahuja@princeton.edu

C. W. RowleyE-mail: cwrowley@princeton.edu

Z. Ma et al.

actuation to maintain the flow around an equilibrium state of interest, for instance to delay transition to turbu-lence, or control separation on a bluff body. Linear control theory provides efficient tools for the analysis anddesign of feedback controllers. However, a significant challenge is that models for flow control problems areoften very high dimensional, e.g., on the order ofO(105∼9), so large that it becomes computationally infeasibleto apply linear control techniques. To address this issue, model reduction, by which a low-order approximatemodel is obtained, is therefore widely employed.

Several techniques are available for model reduction, many of which involve projection onto a set ofmodes. These may be global eigenmodes of a linearized operator [3], modes determined by proper orthogo-nal decomposition (POD) of a set of data [13], and various variants of POD, such as including shift modes[21]. An approach that is used widely for model reduction of linear systems is balanced truncation [20], andwhile this method is computationally intractable for systems with very large state spaces (dimension � 105),recently an algorithm for computing approximate balanced truncation from snapshots of linearized and adjointsimulations has been developed [23] and successfully applied to a variety of high-dimensional flow controlproblems [1,4,14]. In this method, sometimes called balanced POD, one obtains two sets of modes (primaland adjoint) that are bi-orthogonal, and uses those for projection of the governing equations, just as in standardPOD. Compared to most other methods, including POD, balanced truncation has key advantages, such as apriori error bounds, and guaranteed stability of the reduced-order model (if the original high-order systemis stable). Balanced POD is an approximation of exact balanced truncation that is computationally tractablewhen the number of states is very large (for instance, up to 107), and typically produces models that are farmore accurate than standard POD models. For instance, for a linearized channel flow investigated in [14], eventhough the first 5 POD modes capture over 99.7% of the energy in a dataset exhibiting large transient growth,a low-dimensional model obtained by projection onto these modes completely misses the transient growth. Bycontrast, a 3-mode balanced POD model captures the transient growth nearly perfectly; to do as well with astandard POD model, 17 modes were required.

The main steps of balanced POD include (a) taking snapshots from impulse responses of the linearized andadjoint systems, (b) computing a singular value decomposition (SVD) of a matrix formed from inner productsof these snapshots, (c) constructing primal modes and adjoint modes from the resulting singular vectors, and(d) projecting the high-dimensional dynamics onto these modes.

While effective in many examples, balanced POD also faces challenges, especially for use with experi-mental data. The main restriction is that balanced POD requires snapshots of impulse-response data from anadjoint system, and adjoint information is not available for experiments.

To address this issue, here we describe an algorithm widely used for system identification and model reduc-tion, the eigensystem realization algorithm (ERA) [15]. This algorithm has been used for problems in fluidmechanics, primarily as a system-identification technique for flow control [5,6], but also for model reduction[11,24]. Our main result, presented in Sect. 2, is that, for linear systems, ERA theoretically produces exactlythe same reduced-order models as balanced POD, with no need of an adjoint system, and at an order ofmagnitude lower computational cost. This result implies that one can realize approximate balanced truncationeven in experiments, and can also improve computational efficiency in simulations. We note that ERA andsnapshot-based approximate balanced truncation have been applied together in a model reduction procedurein [10]. However, the theoretical equivalence between these two algorithms was not explored in that work.

We present a comparison between balanced POD and ERA, and show that if adjoint information is avail-able, balanced POD also has its own advantages. In particular, balanced POD provides sets of bi-orthogonalprimal/adjoint modes for the linear system, and can be directly generalized to unstable systems. In Sect. 3, wediscuss a modified ERA algorithm that, in the absence of adjoint simulations, uses “pseudo-adjoint modes”to compute reduced-order models; however, this method does not produce balanced models, and performsworse than balanced POD in examples. In Sect. 4, we illustrate these methods using a numerical example ofthe two-dimensional flow past an inclined plate, at a low Reynolds number.

2 The eigensystem realization algorithm as snapshot-based approximate balanced truncation

In this section, we summarize the steps involved in approximate balanced truncation (balanced POD), and theEigensystem Realization Algorithm, and show that they are equivalent.

Balanced truncation involves first constructing a coordinate transformation that “balances” a linear input–output system, in the sense that certain measures of controllability and observability (the Gramian matrices)become diagonal and identical [20]. A reduced-order model is then obtained by truncating the least control-lable and observable states, which correspond to the smallest diagonal entries in the transformed system.

Reduced-order models using eigensystem realization algorithm

Unfortunately, the exact balanced truncation algorithm is not tractable for the large state dimensions encoun-tered in fluid mechanics. However, an approximate, snapshot-based balanced truncation algorithm, referredto as Balanced Proper Orthogonal Decomposition (balanced POD) was proposed in [23], and has been usedsuccessfully in several examples [1,4,14].

The second technique, the ERA, has been used both for system identification and for model reduction, andit is well known that the models produced by ERA are approximately balanced [12,16]. Here we show furtherthat, theoretically, ERA produces exactly the same reduced-order models as balanced POD. This equivalenceindicates that ERA can be regarded as an approximate balanced truncation method, in the sense that, beforetruncation, it implicitly realizes a coordinate transformation under which a pair of approximate controllabilityand observability Gramians are exactly balanced. This feature distinguishes ERA from other model reduc-tion methods that first realize truncations and then balance the reduced-order models. Note that in ERA theGramians, and the balancing transformation itself, are never explicitly calculated, as we will also show in thefollowing discussions.

For both techniques, we will consider a high-dimensional, stable, discrete-time linear system, describedby

x(k + 1) = Ax(k) + Bu(k)

y(k) = Cx(k), (1)

where k ∈ Z is the time step index, u(k) ∈ Rp denotes a vector of inputs (for instance, actuators or dis-

turbances), y(k) ∈ Rq a vector of outputs (for instance, sensor measurements, or simply quantities that one

wishes to model), and x(k) ∈ Rn denotes the state variable (for instance, flow variables at all gridpoints of a

simulation). These equations may arise, for instance, by discretizing the Navier-Stokes equations in time andspace, and linearizing about a steady solution, as will be demonstrated in the example in Sect. 4. The goal isto obtain an approximate model that captures the same relationship between inputs u and outputs y, but witha much smaller state dimension:

xr(k + 1) = Arxr(k) + Bru(k)

y(k) = Crxr(k) (2)

where the reduced state variable xr(k) ∈ Rr , r � n. We consider the discrete-time setting, because we are

primarily interested in discrete-time data from simulations or experiments.

2.1 Snapshot-based approximate balanced truncation (balanced POD)

Here, we give only a brief overview of the balanced POD algorithm, and for details of the method, we referthe reader to [23]. The algorithm involves three main steps:

Step 1: Collect snapshots. Run impulse-response simulations of the primal system (1) and collect mc + 1snapshots of states x(k) in mcP + 1 steps:

X = [B AP B A2P B · · · AmcP B], (3)

where P is the sampling period. In addition, run impulse-response simulations for the adjoint system

z(k + 1) = A∗z(k) + C∗v(k) (4)

where the asterisk ∗ stands for adjoint of a matrix, and collect mo + 1 snapshots of states z(k) inmoP + 1 steps:

Y = [C∗ (A∗)P C∗ (A∗)2P C∗ · · · (A∗)moP C∗]. (5)

Calculate the generalized Hankel matrix,

H = Y ∗X. (6)

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Step 2: Compute modes. Compute the singular value decomposition of H :

H = U�V ∗ = [U1 U2][

�1 00 0

] [V ∗

1V ∗

2

]= U1�1V

∗1 (7)

where the diagonal matrix �1 ∈ Rn1×n1 is invertible and includes all non-zero singular values of

H, n1 = rank(H), and U∗1 U1 = V ∗

1 V1 = In1×n1 . Choose r ≤ n1. Let Ur and Vr denote the sub-matrices of U1 and V1 that include their first r columns, and �r the first r × r diagonal block of �1.Calculate

�r = XVr�− 1

2r ; �r = YUr�

− 12

r . (8)

where the columns of �r and �r are, respectively, the first r primal and adjoint modes of system (1).The two sets of modes are bi-orthogonal: �∗

r �r = Ir×r .Step 3: Project dynamics. The system matrices in the reduced-order model (2) are

Ar = �∗r A�r ; Br = �∗

r B; Cr = C�r. (9)

Note that the n×n controllability/observability Gramians are approximated by the matrices XX∗ and YY ∗.The reduced-order model (2) is obtained by considering a subspace x = �rxr , and projecting the dynamics (1)onto this subspace using the adjoint modes given by �r . It was shown in [23] that �r and �r , respectively, formthe first r columns of the balancing transformation/inverse transformation that exactly balance the approximatecontrollability/observability Gramians XX∗ and YY ∗; see more discussion in Sect. 3.

2.2 The eigensystem realization algorithm

The ERA was proposed in [15] as a system identification and model reduction technique for linear systems.The algorithm follows three main steps [15,16]:

Step 1: Run impulse-response simulations/experiments of the system (1) for (mc +mo)P +2 steps, where mc

and mo, respectively, reflect how much effect is taken for considering controllability and observability,and P again is the sampling period. Collect the snapshots of the outputs y in the following pattern:

(CB, CAB, CAP B, CAP+1B, . . .

CAmcP B, CAmcP+1B, . . . CA(mc+mo)P B, CA(mc+mo )P+1B). (10)

The terms CAkB are commonly called Markov parameters. Construct a generalized Hankel matrixH ∈ R

q(mo+1)×p(mc+1)

H =

⎡⎢⎢⎢⎣

CB CAP B · · · CAmcP B

CAP B CA2P B · · · CA(mc+1)P B...

.... . .

...

CAmoP B CA(mo+1)P B · · · CA(mc+mo)P B

⎤⎥⎥⎥⎦. (11)

Step 2: Compute SVD of H , exactly as in (7), to obtain U1, V1, �1. Let r ≤ rank(H). Let Ur and Vr denotethe sub-matrices of U1 and V1 that include their first r columns, and �r the first r × r diagonal blockof �1.

Step 3: The reduced Ar, Br and Cr in (2) are then defined as

Ar = �− 1

2r U∗

r H ′Vr�− 1

2r ;

Br = the first p columns of �12r V ∗

r ; (12)

Cr = the first q rows of Ur�12r

Reduced-order models using eigensystem realization algorithm

where

H ′ =⎡⎢⎣

CAB CAP+1B · · · CAmcP+1B...

.... . .

...

CAmoP+1B CA(mo+1)P+1B · · · CA(mc+mo)P+1B

⎤⎥⎦ , (13)

which can again be constructed directly from the collected snapshots (10).

2.3 Theoretical equivalence between ERA and balanced POD

The main result of this article is summarized in the following proposition.

Proposition The reduced system matrices Ar, Br and Cr generated in balanced POD and ERA, respectively,by (9) and (12), are theoretically identical.

Proof The first observation is that, with X and Y given by (3) and (5), the generalized Hankel matrices obtainedin balanced POD and ERA, respectively, by (6) and (11), are theoretically identical. The theoretical equivalencebetween the two algorithms then follows: First, H ′ given in (13) satisfies H ′ = Y ∗AX, which, together withthe relation (8), implies the matrices Ar obtained in the two algorithms are identical. To show the equivalence

of Br , first note that when left-multiplied by �− 1

21 U∗

1 on both sides, SVD (7) leads to �− 1

21 U∗

1 H = �121 V ∗

1 .

By definition of Ur, Vr, �r , it implies �− 1

2r U∗

r H = �12r V ∗

r . (Note that it does not imply H = Ur�rV∗r , since

UrU∗r is not the identity.) Thus, in balanced POD,

Br = �∗r B = �

− 12

r U∗r Y ∗B = �

− 12

r U∗r

⎡⎢⎢⎢⎣

CB

CAP B...

CAmoP B

⎤⎥⎥⎥⎦,

which equals the first p columns of �− 1

2r U∗

r H = �12r V ∗

r , which is the matrix Br given by ERA. Sim-

ilarly, the SVD (7) leads to HV1�− 1

21 = U1�

121 and then HVr�

− 12

r = Ur�12r . Thus, in balanced POD,

Cr = C�r = CXVr�− 1

2r , which equals the first q rows of HVr�

− 12

r = Ur�12r , the matrix Cr given by ERA.

��In practice, these two algorithms may generate slightly different reduced-order models, because the Hankel

matrices calculated in the two algorithms are usually not exactly the same, due to small numerical inaccura-cies in adjoint simulations, and/or in matrix multiplications needed to compute the sub-blocks in the Hankelmatrices. In the following discussions, we compare these two algorithms in more detail.

2.4 Comparison between ERA and balanced POD

While ERA and balanced POD produce theoretically identical reduced-order models, the techniques differin several important ways, both conceptually and computationally. Neither ERA nor balanced POD calculateGramians explicitly, but balanced POD does construct approximate controllability and observability matricesX and Y ∗, from which one calculates the generalized Hankel matrix H and balancing transformation. Bal-anced POD thus incurs additional computational cost, because one needs to construct the adjoint system (4),run adjoint simulations for Y , and then calculate each block of H by matrix multiplication. Thus we see thatthe advantages of ERA include:

1. Adjoint-free: ERA is a feasible balanced truncation method for experiments, since it needs only the outputmeasurements from the response to an impulsive input. Note that ERA has been successfully applied inseveral flow control experiments [5,6], as a system-identification technique rather than a balanced-trunca-tion method. In practice, input–output sensor responses are often collected by applying a broadband signalto the inputs, and the ARMARKOV method [2,18] can then be used to identify the Markov parameters,or even directly the generalized Hankel matrix, from the input–output data history.

Z. Ma et al.

Table 1 Comparison of the computational times required for various steps of the algorithms using balanced POD and ERA

Steps in computing reduced-order models Approximate time (CPU hours)

Balanced POD ERA

1. Linearized impulse response 2 42. Computation of POD modes 2 23. Adjoint impulse responses 30 −

(10 in number)4. Computation of the Hankel matrix 7 0.25. Singular value decomposition 0.05 0.056. Computation of modes 1 −7. Computation of models 0.02 0.02

The times are given for a 10-mode output projected system. The Hankel matrices are constructed using 200 state-snapshots fromeach linearized and adjoint simulations for balanced POD, and 400 Markov parameters (outputs) for ERA

2. Computational efficiency: For large problems, typically the most computationally expensive componentof computing balanced POD is constructing the generalized Hankel matrix H in (6), as this involvescomputing inner products of all of the (large) primal and adjoint snapshots with each other. ERA is sig-nificantly more efficient at constructing the matrix H in (11), since only the first row and last column ofblock matrices, i.e., the (mc + mo + 1) Markov parameters, need to be obtained by matrix multiplication.All the other mc × mo block matrices in H are copies of other blocks, and need not be recomputed. Forbalanced POD, the matrix H is obtained by computing all the (mc + 1)× (mo + 1) matrix multiplications(inner products) between corresponding blocks in Y ∗ and X in (6). Thus, for example, if mc = m0 = 200,the computing time needed for constructing H in ERA will be about only 1% of that in balanced POD.See Table 1 for a detailed comparison on computational efficiency between balanced POD and ERA in theexample of the flow past an inclined flat plate.

At the same time, balanced POD also provides its own advantages:

1. Sets of bi-orthogonal primal/adjoint modes: Balanced POD provides sets of bi-orthogonal primal/adjointmodes, the columns of �r and �r . In comparison, without the adjoint system, ERA cannot provide theprimal and adjoint modes. At best, the primal modes may be computed, using the first equation in (8), if thematrix X (3) is stored (in addition to the Markov parameters). But the adjoint modes cannot be computedwithout solutions of the adjoint system. In this sense, balanced POD incorporates more of the physicsof the system (the two sets of bi-orthogonal modes), while ERA is purely based on input–output data ofthe system. The primal/adjoint modes together can be useful for system analysis and controller/observerdesign purposes in several ways: for instance, in flow control applications, a large-amplitude region fromthe most observable mode (the leading adjoint mode) can be a good location for actuator placement. Also,although balanced POD is a linear method, a nonlinear system can be projected onto these sets of modesto obtain a nonlinear low-dimensional model. For instance, the transformation x = �rxr, xr = �∗

r xcan be employed to reduce a full-dimensional nonlinear model x = f (x) to a low-dimensional systemxr = �∗

r f (�rxr). Finally, if parameters (such as Reynolds number or Mach number) are present in theoriginal equations, balanced POD can retain these parameters in the reduced-order models. When thevalues of parameters change, the reduced-order model by balanced POD may still be valid and performwell; see [14] for an application to linearized channel flow.

2. Unstable systems: Balanced POD has been extended to neutrally stable [19] and unstable systems [1]. Inthose cases, one first calculates the right/left eigenvectors corresponding to the neutral/unstable eigenvaluesof the state-transition matrix A, using direct/adjoint simulations. Using these eigenvectors, the system isprojected onto a stable subspace and then balanced truncation is realized for the stable subsystem. ERAfor unstable systems is still an open problem, if adjoint operators are not available. However, we note that,once the stable subsystem is obtained, ERA can still be applied to it and efficiently realize its approximatebalanced truncation.

ERA for systems with high-dimensional outputs. The method of output projection proposed in [23] makes itcomputationally feasible to realize approximate balanced truncation for systems with high-dimensional out-puts—for instance, if one wishes to model the entire state x, say the flow field in the entire computationalor experimental domain. This method involves projecting the outputs onto a small number of POD modes,determined from snapshots of y from the impulse-response dataset. This method can be directly incorporated

Reduced-order models using eigensystem realization algorithm

into ERA as follows: First, run impulse response simulations of the original system and collect Markov param-eters as usual. Then, compute the leading POD modes of the dataset of Markov parameters and stack them ascolumns of a matrix �. Left multiply those Markov parameters by �∗ to project the outputs onto these PODmodes. A generalized Hankel matrix is then constructed using these modified Markov parameters, and theusual steps of ERA follow.

3 A modified ERA method using pseudo-adjoint modes

We have seen that one of the drawbacks of ERA is that it does not provide modes that could be used, forinstance, for projection of nonlinear dynamics, or to retain parameters in the models. More precisely, usingERA, one may still obtain primal modes �1 = XV1�

−1/21 as in balanced POD (see (7–8)), as long as the state

snapshots are collected and stored in X. But it is not possible to obtain the corresponding adjoint modes �1necessary for projection, without performing adjoint simulations to gather snapshots for the matrix Y . This isa severe drawback, as adjoint solutions can be expensive to perform, and are not available for experimentaldata. One idea, proposed in [22], is to define a set of approximate adjoint modes using the Moore-Penrosepseudo-inverse of �1:

�1 = �1(�∗1�1)

−1. (14)

We will call the adjoint modes as defined above the pseudo-adjoint modes corresponding to the modes �1.The system matrices of a r-dimensional reduced-order model (r ≤ rank(H)) generated by this approach thenread

Ar = �∗r A�r ; Br = �∗

r B; Cr = C�r, (15)

where �r and �r are, respectively, the first n × r sub-blocks of �1, �1.While this idea does produce a set of modes that can be used for projection, we show in this section that,

unfortunately, the resulting transformation is not a balancing transformation, and does not produce modelsthat are an approximation to balanced truncation. In fact, the resulting models are closer to those produced bythe the standard POD/Galerkin method: as with standard POD/Galerkin, the method performs well as long asthe most controllable and most observable directions coincide. However, when these directions differ (as isthe case for many problems of interest, including the example in Sect. 4), the method performs poorly. Thesesystems in which controllable and observable directions do not coincide are precisely the systems for whichbalanced POD and ERA give the most improvement over the more traditional POD/Galerkin approach.

3.1 Transformed approximate Gramians

First, let us recall in what sense the model-reduction procedures described in Sect. 2 are approximations tobalanced truncation. Suppose that we have an approximation of the controllability and observability Gramians,factored as

Wc = XX∗, Wo = YY ∗, (16)

where X and Y are the matrices of snapshots from (3) and (5). In balanced POD, we define the primal modes as

columns of �1 = XV1�− 1

21 , and the adjoint modes as columns of �1 = YU1�

− 12

1 , where U1, V1, and �1 aredefined in (7). We will assume in this section that the number of columns of X and Y (the number of snapshots,mc and mo, respectively) is smaller than the number of rows (the state dimension, n), which is always true forthe large fluids systems of interest here.

Then balanced POD is an approximation to balanced truncation in the following sense: as shown in theappendix of [23] (the proof of Proposition 2), we may construct a full (invertible, n × n) transformation

T = [�1 �2] (17)

by choosing �2 such that �∗1 �2 = 0. That is, columns of �2 are orthogonal to the adjoint modes, which are

columns of �1. The inverse transformation then has the form

T −1 =[

�∗1

�∗2

](18)

Z. Ma et al.

where �1 is the matrix of adjoint modes, and �2 is defined by (18). Then, Proposition 2 of [23] states that thetransformed approximate Gramians (16) have the form

T −1Wc(T−1)∗ =

[�1 00 M1

], T ∗WoT =

[�1 00 M2

], (19)

and furthermore the product of the approximate Gramians, in the transformed coordinates, is

T −1WcWoT =[

�21 0

0 0

]. (20)

In this sense, the transformation T balances the approximate Gramians as closely as possible: the Gramiansare block diagonal, and the upper-left blocks are equal and diagonal. Furthermore, all of the states in thelower-right block (i.e., involving M1 and M2 above) are either unobservable or uncontrollable, as they do notappear in the product of the Gramians.

However, if the pseudo-adjoint modes �1 are used in place of the true adjoint modes �1, then this resultdoes not hold, as we now show. Note that, in order for the first block of rows of T −1 to equal �∗

1 , we mustnow define

T = [�1 �2] (21)

where �∗1 �2 = 0. Since the range of �1 equals the range of �1, this is then equivalent to choosing �2 such

that its columns are orthogonal to the columns of �1 (the primal modes), while when the “true” adjoint modesare used, columns of �2 are chosen to be orthogonal to the adjoint modes �1.

Defining �2 by

T −1 =[

�∗1

�∗2

], (22)

one can then show that, as long as rank(X) ≤ rank(Y ),1 the transformed Gramians have the form

T −1Wc(T−1)∗ =

[�1 00 M1

], T ∗WoT =

[�1 M3

M∗3 M2

], T −1WcWoT =

[�2

1 �1M3

M1M∗3 0

], (23)

with

M3 = �1�∗1 �2, (24)

where �1 = YU1�−1/21 are the true adjoint modes. Note that, when the true adjoint modes are used to define

the inverse (18), then M3 = 0, since �∗1 �2 = 0. However, when pseudo-adjoint modes are used, then M3 is

no longer zero, and in fact, can be quite large.An example is shown in Fig. 1, which shows the magnitude of the elements of the transformed Gramians,

where X and Y in (16) are chosen at random. Note that when true adjoint modes are used, the transformedGramians are equal and diagonal, while when the pseudo-adjoint modes are used, the off-diagonal blocks ofthe transformed observability Gramian, and the product of the Gramians have significant magnitude.

Thus, when pseudo-adjoint modes are used, the resulting transformation is not, in general, a balancingtransformation: even though the upper-left blocks of the transformed Gramians are still equal and diagonal,the transformed observability Gramian is not block diagonal, and so its eigenvalues and eigenvectors do notcorrespond to those of the transformed controllability Gramian. Note that this is the whole point of balancedtruncation: to transform to coordinates in which the most controllable directions (dominant eigenvectors of Wc)correspond to the most observable directions (dominant eigenvectors of Wo). Therefore, while the approximatebalanced truncation procedure described in Sect. 2.1 exactly balances the approximate Gramians, transformingby pseudo-adjoint modes does not represent balancing in any meaningful sense.

Note that the matrix M3 describes the degree to which projection using pseudo-adjoint modes fails tobalance the approximate Gramians. This matrix equals zero if the adjoint modes (columns of �1) are spannedby the primal modes (columns of �1). However, M3 is the largest when the dominant adjoint modes (columns

1 If rank(X) > rank(Y ), then the situation is worse, and the transformed controllability Gramian is not block diagonal, nordoes its upper-left block equal �1.

Reduced-order models using eigensystem realization algorithm

50 100 150 200

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200

10

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(a)

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200

10

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30

40

50

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(b)

Fig. 1 Transformed Gramian matrices: (a) using true adjoint modes (Eqs. 19–20) and (b) using pseudo-adjoint modes (Eq. 23).Here, X and Y in (16) are random matrices with n = 200 states and mc = mo = 50 snapshots

of �1) are nearly orthogonal to the dominant primal modes (columns of �1). Unfortunately, this is the case inmany problems of interest, in particular those involving non-normality: the directions spanned by the primalmodes often do not coincide with the directions spanned by the adjoint modes.

In the next section, we apply this approach to the flow past a flat plate, and compare it to the methodsdescribed in Sect. 2.

4 Example: flow past an inclined flat plate

In this section, we illustrate the application of ERA as an approximate balanced truncation method using anumerical example, by obtaining reduced-order models of a large-dimensional fluid system. We compare theresulting models with those obtained using the balanced POD method of [23], ERA with pseudo-adjoint modesas described in Sect. 3, and the standard POD/Galerkin method [13].

4.1 Model problem and parameters

The model problem that we consider is a two-dimensional uniform flow over a flat plate inclined at an angleα = 25◦, at a low Reynolds number Re = 100. At these conditions, the flow asymptotically reaches a stablesteady state, the streamlines of which are plotted in Fig. 2. The numerical method used for all computations isa fast formulation of the immersed boundary method developed by [9], and solves for the vorticity field at eachtime step. We treat farfield boundary conditions using the multiple-grid scheme described in [9] (Sect. 4) withfive nested grids, each with 250 × 250 points. The finest grid covers the region [−2, 3] × [−2.5, 2.5], and thelargest grid covers the region [−32, 48] × [−40, 40], where lengths are nondimensionalized by the chord ofthe flat plate, whose center is located at the origin. The time step used for all simulations is 0.01 (nondimen-sionalized by chord and freestream velocity). The numerical model is the same as that considered in [1] wherebalanced POD is applied for feedback controller design to stabilize an unstable steady state corresponding toa high angle of attack. However, here we consider the case of a stable steady state (with an angle of attack at25◦), for comparison of reduced-order models.

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Fig. 2 Streamlines of the stable steady state past a flat plate at α = 25◦ (left), and the contour-lines of the vorticity field obtainedfrom an impulsive input to the actuator (right)

4.2 Input and output

The governing equations are first linearized about the stable steady state, resulting in a high-dimensional modelin the form of Eq. 1, where the state x consists of the discrete vorticity field at the grid points. See [1] forthe details of the linearized (and adjoint) equations and their numerical formulations. The system input u isa disturbance (or actuator) shown in Fig. 2, modeled as a localized body force in the vicinity of the leadingedge. We consider the output to be the entire velocity field: this is important for capturing the flow physics,and is often needed to represent cost functions used in optimal control design. Since the output is very high-dimensional, in ERA and balanced POD reduction procedures we use output projection described at the endof Sect. 2, projecting the velocity field onto the leading POD modes of the velocity snapshots obtained fromthe impulse response simulation.

4.3 Reduced-order models

ERA is applied to the full-dimensional linearized system to construct a reduced-order model. With a samplingperiod of 50 time steps, 400 adjacent pairs of Markov parameters, as in (10), are collected from an impulseresponse simulation. Since these parameters are a projection of the velocity fields onto the leading POD modes,for an output projection of order m, the number of inner products required is 4m × 102 for construction ofeach H and H ′ (see Sect. 2.4).

For comparison, balanced POD is also used to compute the same reduced-order models. Adjoint simula-tions are performed with the POD modes as initial conditions to compute the matrix Y of (5). The matrices Xand Y are assembled by stacking 200 snapshots from the linearized and each of the adjoint simulations, and inturn, the generalized Hankel matrix H = Y ∗X is computed. For an output projection of order m, the numberof inner products required to compute H is 4m × 104, which is 50 times more than that to compute H and H ′in total for ERA.

We also compare reduced-order models using standard POD modes, and ERA with pseudo-adjoint modes,as described in Sect. 3. The first 100 primal modes are used to compute the pseudo-adjoint modes.

For the given case, a comparison between the computational cost using ERA and using balanced POD isshown in Table 1. Results verify that ERA substantially improves computational efficiency in forming reduced-order models.

Next, we compare the reduced-order models. Figure 3 shows the leading two primal modes and true adjointmodes from balanced POD, compared with the leading two pseudo-adjoint modes. The pseudo-adjoint modeslook quite different from the true adjoint modes, and the flow structures actually more closely resemble theleading primal modes. This result is not surprising, since the pseudo-adjoint modes are always linear combi-nations of the snapshots from the primal simulations, while the true adjoint modes are linear combinations ofsnapshots from adjoint simulations. Following the discussion in the last section, the poor approximation ofthe adjoint modes suggests that the pseudo-adjoint modes may produce poor reduced-order models for thisexample, as we will verify below.

Figure 4 shows the diagonal values of the controllability and observability Gramians, as well as the empir-ical Hankel singular values, for reduced-order models obtained from three different methods: ERA, balancedPOD, and ERA with pseudo-adjoint modes. The models obtained using ERA are more accurate in the sensethat the three sets of curves are almost indistinguishable, for all orders of output projection. However, forbalanced POD, the diagonal values of the observability Gramians are accurate only for certain leading modes,

Reduced-order models using eigensystem realization algorithm

Fig. 3 The first two primal and adjoint modes computed by using balanced POD, and the first two pseudo-adjoint modes computedby using (14) and the first 100 primal modes. Modes are illustrated using contour plots of the vorticity field

the number of which depends on and increases with the order of output projection. This inaccuracy can beattributed to a slight inaccuracy in the adjoint formulation, which in turn results from an approximation in themulti-domain approach used to treat farfield boundary conditions in the immersed boundary method of [9];see [1] for more details. Thus, ERA is advantageous as it does not need any adjoint simulations and resultsin more balanced Gramians. On the other hand, ERA with pseudo-adjoint modes generates poorly balancedcontrollability and observability Gramians, as shown in Fig. 4c. This is because the leading primal modes andadjoint modes are supported very differently in the spatial domain, and thus the pseudo-adjoint modes, basedon linear combination of leading primal modes, poorly approximate the true adjoint modes.

4.4 Model performance

We can quantify the performance of the various reduced-order models by computing error norms. One suchmeasure is the 2-norm of the error between the impulse response of the full linearized system, denoted G(t),and that of a reduced-order model with r modes, denoted by Gr(t). We first compute the 2-norm of the errorbetween the full system (with the entire velocity field as output) and the output-projected system of order20, shown as the horizontal dashed line in Fig. 5. This is the lower error bound for any reduced-order modelof the given output-projected system. Results shown in Fig. 5 indicate that the first several low-order modelsobtained by ERA and balanced POD generate slightly different 2-norms of error, presumably because of theslight inaccuracy in the adjoint, mentioned previously. For most orders, however, they agree; and both errornorms converge to the lower bound as the order of the model increases. By running more simulation tests, weobserve that with higher-order output projections, ERA and balanced POD error norms converge to each otherfaster when the order of model increases.

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(a) (b)

(c)

Fig. 4 Comparison of Gramians computed using (a) ERA, (b) balanced POD, and (c) ERA with pseudo-adjoint modes: Theempirical Hankel singular values (solid line) and the diagonal elements of the controllability (circle) and observability (times)Gramians with different order of modes (e.g., 4, 10, 20) in output projection

Fig. 5 H2−norm of the error with increasing order of the reduced-order models: exact output of the output-projected system(dashed line); models obtained using balanced POD (square), ERA (times), ERA with pseudo-adjoint modes (down-pointingtriangle), and POD (up-pointing triangle). A 20-mode output projection is used in ERA, balanced POD, and ERA with pseudo-adjoint modes

Figure 5 also shows the 2-norm error plots for models by ERA with pseudo-adjoint modes, using 20-modeoutput projection, and for models computed using standard POD. Errors of models by ERA with pseudo-adjoint modes converge to the lower bound much slower than ERA/balanced POD. Errors of models by PODdo not start converging until more than nearly 20 modes are used, and they converge to a larger error boundthan ERA/balanced POD, again because POD models do not capture the input–output dynamics as well asbalanced truncation based models.

Reduced-order models using eigensystem realization algorithm

Fig. 6 The first output, output a1, from the impulse-response simulation: results of full-simulation (circle), compared with thoseof 16-mode reduced order model by ERA (times), 30-mode model by ERA with pseudo-adjoint modes (down-pointing triangle),and 30-mode model by POD (up-pointing triangle). A 20-mode output projection is used in ERA and ERA with pseudo-adjointmodes

frequency (Hz)

mag

nitu

de (

dB)

Fig. 7 Singular-value plots: The full system and 30-mode models obtained using balanced POD, ERA, ERA with pseudo-adjointmodes, and POD, all with a 20-mode output projection. ERA and balanced POD models generate almost identical plots

In the time domain, a comparison of the transient response to an impulsive disturbance is shown in Fig. 6, inwhich the first output of the reduced-order model is plotted, for a 16-mode model determined by ERA, and for30-mode models by POD and ERA with pseudo-adjoint modes. The 16-mode ERA model already accuratelypredicts the response for all times. The higher-dimensional, 30-mode models using POD and pseudo-adjointmodes are both stable, and perform reasonably well; however, they overpredict the response, particularly aftertime t ≈ 80.

We also compare the frequency response of reduced-order models to that of the full system, or more pre-cisely, the full output-projected system. One way to represent the response of a single-input multiple-outputsystem is by a singular-value plot, a plot of the maximum singular value of the transfer function matrix as afunction of frequency. To generate this plot, a very long simulation of 5 × 105 time steps for the full system isperformed, with a random input sampled from a uniform distribution in the range (−0.5, 0.5). The output snap-shots are projected onto leading POD modes. The magnitude of the transfer function is then computed from thecross spectrum of the input and outputs (using the Matlab command tfestimate). Finally, singular-valueplots for the full output-projected systems are obtained, with a typical case shown in Fig. 7.

A typical set of singular-value plots of different reduced-order models are presented in Fig. 7. Resultsshown in the figure indicate that ERA and balanced POD 30-mode models are almost identical, and are closeto the corresponding full output-projected system. In comparison, Fig. 7 also shows singular-value plots for30-mode models by ERA with pseudo-adjoint modes and by POD. Note that for computational feasibility,here the output of the POD model is the first twenty reduced states, i.e., the full-dimensional output of the PODmodel are projected onto the leading twenty POD modes. The frequency responses of the models by POD andERA with pseudo-adjoint modes capture the resonant peak, but do not match well for frequencies far awayfrom the resonant peak. These two models both generate spurious peaks in the frequency range of [0.1, 2].

5 Discussion

We report that, theoretically, the ERA and snapshot-based approximate balanced truncation (balanced POD)produce exactly the same reduced-order models. This equivalence implies that ERA balances a pair of approxi-

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mate Gramians and thus can be regarded as an approximate balanced truncation method. Compared to balancedPOD, the main features of ERA are that it does not require data from adjoint systems and therefore can beused with experimental data; furthermore, its construction of the generalized Hankel matrix is computation-ally an order-of-magnitude cheaper than balanced POD. Numerical results indicate that ERA can be moreaccurate than balanced POD in practice, since there can be slight inaccuracies in the adjoint operator usedwith balanced POD. Balanced POD does have its own advantages, however: unlike ERA, it produces sets ofbi-orthogonal modes that are useful for other purposes. Nonlinear models may be obtained by projection ontothese modes; and parameters such as Reynolds number can be retained in the reduced-order models generatedusing these modes. Balanced POD has also been generalized for unstable systems. We also examine a mod-ified ERA approach in which one constructs sets of bi-orthogonal modes without using adjoint information,using a matrix pseudo-inverse, as in [22]. Although this approach provides sets of bi-orthogonal modes (pri-mal/pseudo-adjoint modes), in general it cannot be regarded as an approximate balanced truncation method,since it does not balance the approximate Gramians.

We have demonstrated the methods on a model problem consisting of a disturbance interacting with theflow past an inclined flat plate. As expected, balanced POD models perform nearly identically to ERA models.The small differences result because the adjoint simulation required for balanced POD is not a perfect adjointat the discrete level. Both procedures work significantly better than standard POD models, or ERA modelsusing pseudo-adjoint modes for projection.

Finally, we emphasize that throughout we have considered only stable, linear models. Possible futuredirections of this study include a generalization to unstable systems, and ultimately to nonlinear systems.

Acknowledgments The authors gratefully acknowledge the support by U. S. Air Force Office of Scientific Research grantsFA9550-05-1-0369 and FA9550-07-1-0127. The authors thank Kunihiko Taira for help with the numerical solver.

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